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Volume Issue 60 Jan , pp. Volume Issue 59 Jul , pp. In Sect. These equations, particularly that of Theorem 4. An important foundational step in the study of any system of evolutionary partial differential equations is to show short-time existence and uniqueness. For the Ricci flow, unfortunately, short-time existence does not follow from standard parabolic theory, since the flow is only weakly parabolic. To overcome this, Hamilton's seminal paper [Ham82b] employed the deep Nash —Moser implicit function theorem to prove short-time existence and uni- queness. A detailed exposition of this result and its applications can be found in Hamilton's survey [Ham82a].
DeTurck [DeT83]later found a more direct proof by modifying the flow by a time-dependent change of variables to make it parabolic. It is this method that we will follow. In Theorem 4. The maximum principle is the main tool we will use to understand the behaviourof solutions to the Ricci flow. While other problems arising in geo- metric analysis and calculus of variations make strong use of techniques from functional analysis, here — due to the fact that the metric is changing — most of these techniques are not available; although methods in this direction are developed in the work of Perelman [Per02].
The maximum principle, though very simple, is also a very powerful tool which can be used to show that pointwise inequalities on the initial data of parabolic pde are preserved by the evolution. As we have already seen, when the metric evolves by Ricci flow the various curvature tensors R, Ric, and Scal do indeed satisfy systems of parabolic pde.
Our main applications of the maximum principle will be to prove that certain inequalities on these tensors are preserved by the Ricci flow, so that the geometry of the evolving metrics is controlled. In Chaps. By appealing to this view, we would expect the same kind of regularity that is seen in parabolic equa- tions to apply to the curvature. In particular we want to show that bounds on curvature automatically induce a priori bounds on all derivatives of the curvature for positive times.
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In the literature these are known as Bernstein— Bando—Shi derivative estimates as they follow the strategy and techniques introduced by Bernstein done in the early twentieth century for proving gradient bounds via the maximum principle and were derived for the Ricci flow in [Ban87] and comprehensively by Shi in [Shi89]. In the second section we use these bounds to prove long-time existence. Write a customer review. Discover the best of shopping and entertainment with Amazon Prime. Prime members enjoy FREE Delivery on millions of eligible domestic and international items, in addition to exclusive access to movies, TV shows, and more.
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